extreme T(x,y)=ln(3xy+2x^2-y)

解答

极值点

T (x, y) = ln (3 x y + 2 x^{2} - y)

鞍点 (\frac{1}{3}, - \frac{4}{9})

求解步骤

找到临界点:

(\frac{1}{3}, - \frac{4}{9})

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{( 3 x - 1 ) ^{2}}{( 3 x y + 2 x ^{2} - y ) ^{2}}

D (x, y) = \frac{36 x ^{4} + 108 x ^{3} y + 81 x ^{2} y ^{2} - 36 x ^{2} y - 8 x ^{2} - 54 x y ^{2} - 12 x y + 9 y ^{2} + 4 y}{( - y + 2 x ^{2} + 3 x y ) ^{4}}

检查在每个临界点处的符号

检查临界点

(\frac{1}{3}, - \frac{4}{9}) :

鞍点

鞍点 (\frac{1}{3}, - \frac{4}{9})

e x t re m e f (x) = 5000 - 15 x + 0.08 x^{2}

e x t re m e 2 y^{2} + 2 x y + x^{2} - 16 x - 20 y

e x t re m e f (x) = - 2 x^{2} + 200 x

e x t re m e y = x^{2} - 5 x - 14

e x t re m e x \cdot 7 - x